Atomic Structure
Introduction :
STRUCTURE OF ATOM
Rutherford's Model Bohr's Model Wave mechanical model
Dalton’s concept of the indivisibility of the atom was completely discredited by a series of experimental evidences obtained by scientists. A number of new phenomena were brought to light and man’s idea about the natural world underwent a revolutionary change. the discovery of electricity and spectral phenomena opened the door for radical changes in approaches to experimentation. It was concluded that atoms are made of three particles : electrons, protons and neutrons. These particles are called the fundamental particles of matter.
Earlier efforts to reveal structure of atom :
CATHODE RAYS - DISCOVERY OF ELECTRONFigure-1
In 1859 Julius Plucker started the study of conduction of electricity through gases at low pressure (10–4atm) in a discharge tube When a high voltage of the order of 10,000 volts or more was impressed across
the electrodes, some sort of invisible rays moved from the negative electrode to the positive electrode these rays are called as cathode rays.
PROPERTIES OF CATHODE RAYS :
Cathode rays have the following properties.
(i) Path of travelling is straight from the cathode with a very high velocity as it produces shadow of an object placed in its path.
(ii) Cathode rays produce mechanical effects. If small light paddle wheel is placed between the electrodes, it rotates. This indicates that the cathode rays consist of material particles.
(iii) When electric and magnetic fields are applied to the cathode rays in the discharge tube. The rays are deflected thus establishing that they consist of charged particles. The direction of deflection showed that cathode rays consist of negatively charged particles called electrons.
(iv) They produce a green glow when strike the glass wall beyond the anode. Light is emitted when they strike the zinc sulphide screen.
(v) Cathode rays penetrate through thin sheets of aluminium and metals. (vi) They affect the photographic plates
(vii) The ratio of charge(e) to mass(m) i.e. charge/mass is same for all cathode rays irrespective of the gas used in the tube. e/m = 1.76 × 1011Ckg–1
Thus, it can be concluded that electrons are basic constituent of all the atoms.
PRODUCTION OF ANODE RAYS (DISCOVERY OF PROTON) :
Goldstein (1886) repeated the experiment with a discharge tube filled with a perforated cathode and found that new type of rays came out through the hole in the cathode.
Figure-3
When this experiment is conducted, a faint red glow is observed on the wall behind the cathode. Since these rays originate from the anode, they are called anode rays.
PROPERTIES OF ANODE RAYS :
Anode rays travel along straight paths and hence they cast shadows of object placed in t h e i r path.
They rotate a light paddle wheel placed in their path. This shows that anode rays are made up of material particles.
They are deflected towards the negative plate of an electric field. This shows that these rays are positively charged.
For different gases used in the discharge tube, the charge to mass ratio (e/m) of the positive particles constituting the positive rays is different. When hydrogen gas is taken in the discharge tube, the e/ m value obtained for the positive rays is found to be maximum. Since the value of charge (e) on the positive particle obtained from different gases is the same, the value of m must be minimum for the positive particles obtained from hydrogen gas. Thus, the positive particle obtained from hydrogen gas is the lightest among all the positive particles obtained from different gases. This particle is called the proton.
DISCOVERY OF NEUTRON :
Later, a need was felt for the presence of electrically neutral particles as one of the constituent of atom. These particles were discovered by Chadwick in 1932 by bombarding a thin sheet of Beryllium with -particles, when electrically neutral particles having a mass slightly greater than that of the protons were emitted. He named these particles as neutrons.
n C He Be 42 126 10 9 4
The NUCLEUS :
Electrons, protons & neutrons are the fundamental particles present in all atoms,(except hydrogen)
Table : 1
Particles Symbol Mass Charge Discoverer Electron –1e0or 9.10939 x 10-31kg – 1.6022 x l0–19 J.J. Thomson
Coulombs Stoney Lorentz 1887
0.00054 u – 4.803 × 10–10esu
Proton 1H1 1.6722 x 10–27kg + 1.6022 x 10–19 Goldstein
Coulombs Rutherford1907
1.00727 u + 4.803 x 10–10esu
Neutron 0n1 1.67493 x 10–27kg neutral James Chadwick
1.00867 u 0 1932
1 amu 1.66 × 10–27kg
ATOMIC MODELS :
( A ) Thomson’s Model of the Atom :
An atom is electrically neutral. It contains positive charges (due to the presence of protons ) as well as negative charges (due to the presence of electrons). Hence, J.J. Thomson assumed that an atom is a uniform sphere of positive charges with electrons embedded in it.
Figure-4
(B) Rutherford’s Experiment :
Observation :
1. Most of the-particles passed straight through the gold foil without suffering any deflection from their original path.
2. A few of them were deflected through small angles, while a very few were deflected to a large extent. 3. A very small percentage(1 in 20000)was deflected through angles ranging from 90° to 180°.
Rutherford’s nuclear concept of the atom.
(i) The atom of an element consists of a small positively charged ‘nucleus’ which is situated at the centre of the atom and which carries almost the entire mass of the atom.
(ii) The electrons are distributed in the empty space of the atom around the nucleus in different concentric circular paths, called orbits.
(iii) The number of electrons in orbits is equal to the number of positive charges (protons) in the nucleus. Hence, the atom is electrically neutral.
(iv) The volume of the nucleus is negligibly small as compared to the volume of the atom. (v) Most of the space in the atom is empty.
DRAWBACKS OF RUTHERFORD’S MODEL :
1. This was not according to the classical theory of electromagnetism proposed by maxwell. According to this theory, every accelerated charged particle must emit radiations in the form of electromagnetic waves and loses it total energy.
Since energy of electrons keep on decreasing, so radius of the circular orbits should also decrease and ultimately the electron should fall in nucleus.
2. It could not explain the line spectrum of H-atom.
PROPERTIES OF CHARGE :
1. Q = ne ( charge is quantized) 2. Charge are of two types :
(i) positive charge (ii) Negative Charge e = –1.6 x 10–19
p = + 1.6 x 10–19C
This does not mean that a proton has a greater charge but it implies that the charge is equal and opposite. Same charge repel each other and opposite charges attract each other.
3. Charge is a SCALAR Qty. and the force between the charges always acts along the line joining the charges.
The magnitude of the force between the two charge placed at a distance ‘r’ is given by FE = 0 4 1 2 2 1 r q q (electrical force)
4. If two charge q1and q2are sepreated by distance r then the potential energy of the two charge system is given by. P.E. = 0 4 1 r q q1 2
5. If a charged particle q is placed on a surface of potential V then the potential energy of the charge is q x V.
Ex. An-particle is projected from infinity with the velocity V0towards the nucleus of an atom having atomic number equal to Z then find out (i) closest distance of approach (R) (ii) what is the velocity of the-particle at the distance R1(R1> R) from the nucleus.
Sol. + m V 1 3 2 R R1
From energy conservation P.E1+ KE1= P.E2+ KE2
0 + 2 1 mV2= R ) e 2 )( Ze ( K + 0 R = 2 2 V m KZe 4
(closest distance of approch)
Let velocity at R1is V1.
From energy conservation P.E1+ KE1= P.E3+ KE3
0 + 2 1 mV2= 1 R ) e 2 )( Ze ( K + 2 1 mV12
Size of the nucleus :
The volume of the nucleus is very small and is only a minute fraction of the total volume of the atom. Nucleus has a diameter of the order of 10–12to 10–13cm and the atom has a diameter of the order of 10–8cm.
Thus, diameter (size) of the atom is 100,000 times the diameter of the nucleus.
The radius of a nucleus is proportional to the cube root of the number of nucleons within it. R = R0(A)1/3cm
where R0can be 1.1 × 10–13 to 1.44 × 10–13cm ; A = mass number ; R = Radius of the nucleus.
Nucleus contains protons & neutrons except hydrogen atoms which does not contain neutron in the nucleus.
ATOMIC NUMBER (Z) AND MASS NUMBER (A) :
Atomic number (Z) of an element= Total number of protons present in the nucleus = Total number of electrons present in the atom
Atomic number is also known as proton number because the charge on the nucleus depends upon the number of protons.
Since the electrons have negligible mass, the entire mass of the atom is mainly due to protons and neutrons only. Since these particles are present in the nucleus, therefore they are collectively called nucleons.
As each of these particles has one unit mass on the atomic mass scale, therefore the sum of the number of protons and neutrons will be nearly equal to the mass of the atom.
Mass number of an element = No. of protons (Z) + No. of neutrons (n). The mass number of an element is nearly equal to the atomic mass of that element. However, the main difference between the two is that mass number is always a whole number whereas atomic mass is usually not a whole number.
The atomic number (Z) and mass number (A) of an element ‘X’ are usually represented alongwith the symbol of the element as
1. sotopes : Such atoms of the same element having same atomic number but different mass numbers are called isotopes.
D , H12 1 1 and T 3
1 and named as protium, deuterium (D) and tritium (T) respectively. Ordinary hydrogen
is protium.
2. Isobars : Such atoms of different elements which have same mass numbers (and of course different atomic numbers) are called isobars
e.g. 1840Ar,1940K,2040Ca.
3. Isotones : Such atoms of different elements which contain the same number of neutrons are called isotones
e.g. 146 C,157N,168O.
4. Isoelectronic : The species (atoms or ions) containing the same number of electrons are called isoelectronic.
For example, O2–, F–, Na+, Mg2+, Al3+, Ne all contain 10 electrons each and hence they are
isoelectronic.
Ex-1. Complete the following table :
Particle Mass No. Atomic No. Protons Neutrons Electrons
Nitrogen atom – – – 7 7
Calcium ion – 20 – 20 –
Oxygen atom 16 8 – – –
Bromide ion – – – 45 36
Sol. For nitrogen atom.
No. of electron = 7 (given) No. of neutrons = 7 (given) No. of protons = Z = 7 ( atom is electrically neutral) Atomic number = Z = 7
Mass No. (A) = No. of protons + No. of neutrons = 7 + 7 = 14
For calcium ion.
No. of neutrons = 20 (Given) Atomic No. (Z) = 20 (Given) No. of protons = Z = 20 ;
No. of electrons in calcium atom = Z = 20
But in the formation of calcium ion, two electrons are lost from the extranuclear part according to the equation Ca Ca2+ + 2e–but the composition of the nucleus remains unchanged.
No. of electrons in calcium ion = 20 – 2 = 18
Mass number (A) = No. of protons + No. of neutrons = 20 + 20 = 40.
For oxygen atom.
Mass number (A) = No. of protons + No. of neutrons = 16 (Given
Atomic No. (Z) = 8 (Given) No. of protons = Z = 8, No. of electrons = Z = 8
For bromide ion.
No. of neutrons = 45 (given) No. of electrons = 36 (given)
But in the formation of bromide ion, one electron is gained by extra nuclear part according to equation Br + e– Br–, But the composition of nucleus remains unchanged.
No. of protons in bromide ion = No. of electrons in bromine atom = 36 – 1 = 35 Atomic number (Z) = No. of protons = 35
Mass number (A) = No. of neutrons + No. of protons = 45 + 35 = 80.
Electromagnetic wave radiation :
The oscillating electrical/magnetic field are electromagnetic radiations. Experimentally, the direction of oscillations of electrical and magnetic field are prependicular to each other.
E = Electric field,
B = Magnetic field
Figure-6
Direction of wave propogation.
Figure-7
Some important characteristics of a wave :
Figure-8
Wavelength of a wave is defined as the distance between any two consecutive crests or troughs. It is represented by (lambda) and is expressed in Å or m or cm or nm (nanometer) or pm (picometer).
1 Å = 10– 8cm = 10–10m
Frequency of a wave is defined as the number of waves passing through a point in one second. It is represented by (nu) and is expressed in Hertz (Hz) or cycles/sec or simply sec–1or s–1.
1 Hz = 1 cycle/sec
Velocity of a wave is defined as the linear distance travelled by the wave in one second. It is represented by v and is expressed in cm/sec or m/sec (ms–1).
Amplitude of a wave is the height of the crest or the depth of the trough. It is represented by ‘a’ and is expressed in the units of length.
Wave number is defined as the number of waves present in 1 cm length. Evidently, it will be equal to the reciprocal of the wavelength. It is represented by (read as nu bar).
1
If is expressed in cm, will have the units cm–1.
Relationship between velocity, wavelength and frequency of a wave. As frequency is the number of waves passing through a point per second and is the length of each wave, hence their product will give the velocity of the wave. Thus
v = ×
Order of wavelength in Electromagnetic spectrum
Cosmic rays < – rays < X-rays < Ultraviolet rays < Visible < Infrared < Micro waves < Radio waves.
Particle Nature of Electromagnetic Radiation : Planck's Quantum Theory
Some of the experimental phenomenon such as diffraction and interference can be explained by the wave nature of the electromagnetic radiation. However, following are some of the observations which could not be explained
(i) the nature of emission of radiation from hot bodies (black - body radiation)
(ii) ejection of electrons from metal surface when radiation strikes it (photoelectric effect)
Black Body Radiation :
When solids are heated they emit radiation over a wide range of wavelengths.
The ideal body, which emits and absorbs all frequencies, is called a black body and the radiation emitted by such a body is called black body radiation. The exact frequency distribution of the emitted radiation (i.e., intensity versus frequency curve of the radiation) from a black body depends only on its temperature.
Figure-9
The above experimental results cannot be explained satisfactorily on the basis of the wave theory of light. Planck suggested that atoms and molecules could emit (or absorb) energy only in discrete quantities and not in a continuous manner.
QUANTUM THEORY OF LIGHT :
The smallest quantity of energy that can be emitted or absorbed in the form of electromagnetic radiation is called as quantum of light.
According to Planck, the light energy coming out from any source is always an integral multiple of a smallest energy value called quantum of light.
Let quantum of light be = E0(J), then total energy coming out is = nE0 (n = Integer) Quantum of light = Photon ( Packet or bundle of energy)
Energy of one photon is given by E0= h (- Frequency of light)
h = 6.626 x 10–34J-Sec (h - Planck const.)
E0= hc (c - speed of light) ( - wavelength) Order of magnitude of Eo= 10 8 34 10 10 10 = 10–16J
Ex-2. Certain sun glasses having small of AgCl incorporated in the lenses, on expousure to light of appropriate wavelength turns to gray colour to reduce the glare following the reactions:
AgCl hv Ag(Gray) + Cl
If the heat of reaction for the decomposition of AgCl is 248 kJ mol–1, what maximum wavelength is needed to induce the desired process?
Sol. Energy needed to change = 248 × 103J/mol
If photon is used for this purpose, then according to Einstein law one molecule absorbs one photon. Therefore, NA.hc = 248 × 103 = 3 23 8 34 10 248 10 023 . 6 10 0 . 3 10 626 . 6 = 4.83 × 10–7m
One electron volt (e.v.)
: Energy gained by an electron when it is accelerated from rest through a potential difference of 1 volt.Note : Positive charge always moves from high potential to low potential and –ve charge always. moves from low
potential to high potential if set free.
Figure-10
From Energy conservation principle, P.E.i+ K.E.i = P.E.f+ K.E.f
(– e) 0 + 0 = (– e)(1V) + 2 1 2 f mV K.E. = 2 1 2 f mV = e(1 volt)
If a charge ‘q’ is accelerated through a potential dirrerence of ‘V’ volt then its kinetic energy will be increased by q.V.
1eV = 1.6 x 10–19C x 1 volt 1eV = 1.6 x 10–19J
Photoelectric Effect :
When certain metals (for example Potassium, Rubidium, Caesium etc.) were exposed to a beam of light electrons were ejected as shown in Fig.
Figure-11
The phenomenon is called Photoelectric effect. The results observed in this experiment were :
(i) The electrons are ejected from the metal surface as soon as the beam of light strikes the surface, i.e., there is no time lag between the striking of light beam and the ejection of electrons from the metal surface. (ii) The number of electrons ejected is proportional to the intensity or brightness of light.
(iii) For each metal, there is a characteristic minimum frequency,0(also known as threshold frequency) below which photoelectric effect is not observed. At a frequency > 0, the ejected electrons come out with certain kinetic energy. The kinetic energies of these electrons increase with the increase of frequency of the light used.
When a photon of sufficient energy strikes an electron in the atom of the metal, it transfers its energy instantaneously to the electron during the collision and the electron is ejected without any time lag or delay. Greater the energy possessed by the photon, greater will be transfer of energy to the electron and greater the kinetic energy of the ejected electron. In other words, kinetic energy of the ejected electron is proportional to the frequency of the electromagnetic radiation. Since the striking photon has energy equal to h and the minimum energy required to eject the electron is h0(is also called work function, W0) then the difference in energy (h – h0) is transferred as the kinetic energy of the photoelectron. Following the conservation of energy principle, the kinetic energy of the ejected electron is given by the equation
h = h0+ 2 1
me2
where meis the mass of the electron and v is the velocity associated with the ejected electron.
Ex-3. The threshold frequency0for a metal is 6 × 1014s–1. Calculate the kinetic energy of an electron emitted when radiation of frequency = 1.1 × 1015s–1hits the metal.
Sol. K.E. = 2 1 meV2= h ( – 0) K.E. = (6.626 × 10–34) (1.1 × 1015– 6 × 1014) K.E. = (6.626 × 10–34) (5 × 1014) = 3.313 × 10–19J
BOHR’S ATOMIC MODEL :
It is based on quantum theory of light. Assumptions of Bohr’s model : There are certain orbits around the nucleus such that if electron will be revolving in these orbit, then it not emit any electromagnetic radiation. These are called stationary orbit.
The neccessary centripetal force is produced by attraction forces of nucleus.
r mv2 = 2 2 r Z Ke
Angular momentum of the electron in these stationary orbit is always an integral multiple of
2 h
mvr =
2 nh
Electron can make jump from one stationary orbit to another stationary orbit by absorbing or emitting a photon of energy equal to difference in the energies of the stationary orbit.
hc
=E E – difference in the energy of orbit
Mathematical forms of Bohr’s Postulates :
Calculation of the radius of the Bohr’s orbit : Suppose that an electron having mass ‘m’ and charge ‘e’ revolving around the nucleus of charge ‘Ze’ (Z is atomic number & e = charge) with a tangential/linear velocity of ‘v’. Further consider that ‘r’ is the radius of the orbit in which electron is revolving.
According to Coulomb’s law, the electrostatic force of attraction (F) between the moving electron and nucleus is – F = 2 2 r KZe where : K = constant = 0 4 1 = 9 x 109Nm2/C2 and the centrifugal force F =
r mv2
For the stable orbit of an electron both the forces are balanced. i.e r mv2 = 2 2 r KZe then v2= mr KZe2 ... (i) From the postulate of Bohr,
mvr = 2 nh v = mr 2 nh + V On squaring v2= 2 2 2 2 2 r m 4 h n ... (ii)
Figure-12
From equation (i) and (ii)
mr KZe2 = 2 2 2 2 2 r m 4 h n On solving, we will get
r = 2 2 2 2 mKZe 4 h n
On putting the value of e , h , m, the radius of nthBohr orbit is given by :
rn= 0.529 x
Z n2
Ex.4 Calculate radius ratio for 2ndorbit of He+ion & 3rdorbit of Be+++ion. Sol. r1(radius of 2ndorbit of He+ion) = 0.529
2 22 Å
r2(radius of 3rdorbit of Be+++ion) = 0.529
4 32 Å Therefore 2 1 r r = 4 / 3 529 . 0 2 / 2 529 . 0 2 2 = 9 8
Calculation of velocity of an electron in Bohr’s orbit :
Angular momentum of the revolving electron in nthorbit is given by
mvr = 2 nh v = mr 2 nh ... (iii)
put the value of ‘r’ in the equation (iii) then, v = 2 2 2 2 h mn 2 K mZe 4 nh v = nh K Ze2
on putting the values of, e-, h and K
velocity of electron in nth orbit v
n= 2.18 x 106x n
Z
m/sec ; v Z ; v
n 1
T, Time period of revolution of an electron in its orbit = v
r 2
f, Frequency of revolution of an electron in its orbit = r 2
v Calculation of energy of an electron :
The total energy of an electron revolving in a particular orbit is T.E. = K.E. + P.E.
where :
P.E. = Potential energy , K.E. = Kinetic energy , T.E. = Total energy The K.E. of an electron =
2 1
mv2
and the P.E. of an electron = –
r KZe2 Hence, T.E. = 2 1 mv2– r KZe2 we know that, r mv2 = 2 2 r KZe or mv2 = r KZe2
T.E. = r 2 KZe2 – r KZe2 = – r 2 KZe2 So, T.E. = – r 2 KZe2
substituting the value of ‘r’ in the equation of T.E. Then T.E. = – 2 KZe2 x 2 2 2 2 h n m Ze 4 = – 2 2 2 4 2 2 h n K m e Z 2 Thus, the total energy of an electron in nthorbit is given by
T.E. = En= – 2 2 4 2 h k me 2 2 2 n z ... (iv) Putting the value of m,e,h and we get the expression of total energy En = – 13.6 2
2
n Z
eV / atom n T.E. ; Z T.E.
= – 2.18 × 10–18 2 2 n Z J/atom T.E. = 2 1 P.E. T.E. = – K.E.
Note : - The P.E. at the infinite = 0 The K.E. at the infinite = 0 Conclusion from equation of energy :
(a) The negative sign of energy indicates that there is attraction between the negatively charged electron and positively charged nucleus.
(b) All the quantities on R.H.S. in the energy equation [Eq. iv] are constant for an element having atomic number Z except ‘n’ which is an integer such as 1,2,3, etc . i.e. the energy of an electron is constant as long as the value of ‘n’ is kept constant.
(c) The energy of an electron is inversely proportional to the square of ‘n’ with negative sign.
Ex-4 What are the frequency and wavelength of a photon emitted during a transition from n = 5 state to the n = 2 state in the hydrogen atom ?
Sol. Since ni= 5 and nf= 2, this transition gives rise to a spectral line in the visible region of the Balmer series. E = 2.18 × 10–18J 2 2 2 1 – 5 1 = – 4.58 × 10–19J It is an emission energy
The frequency of the photon (taking energy in terms of magnitude) is given by = h E = Js 10 626 . 6 J 10 58 . 4 34 – 19 – = 6.91 × 1014Hz = c = Hz 10 91 . 6 ms 10 0 . 3 14 1 – 8 = 434 nm
Failures / limitations of Bohr’s theory:
(a) He could not explain the line spectra of atoms containing more than one electron. (b) He also could not explain the presence of multiple spectral lines.
field (Stark effect)
(d) No conclusion was given for the principle of quantisation of angular momentum. (e) He was unable to explain the de-Broglie’s concept of dual nature of matter. (f) He could not explain Heisenberg’s uncertainty principle.
Energy Level Diagram :
(i) Orbit of lowest energy is placed at the bottom, and all other orbits are placed above this. (ii) The gap between two orbits is proportional to the energy difference of the orbits.
-0.85 eV -13.6eV -3.4eV -1.51eV n= n=4 n=3 n=2 n=1 10.2eV 12.1eV
Energy level diagram of H-atom
0 eV
Figure-13
DEFINITION VALID FOR SINGLE ELECTRON SYSTEM :
(i) Ground state :
Lowest energy state of any atom or ion is called ground state of the atom It is n = 1. Ground state energy of H–atom = – 13.6 ev
Ground state energy of He+on = – 54.4 ev
(ii) Excited State :
States of atom other than the ground state are called excited states : n = 2 first excited state
n = 3 second excited state n = 4 third excited state n = n + 1 nthexcited state
(iii) Ionisation energy (E) :
Minimum energy required to move an electron from ground state to n = is called ionisation energy of the atom or ion.
onisation energy of H–atom = 13.6 ev onisation energy of He+ion = 54.4 ev
onisation energy of Li+2ion = 122.4 ev
(iv) Ionisation Potential (.P.) :
Potential difference through which a free electron must be accelerated from rest, such that its kinetic energy becomes equal to ionisation energy of the atom is called ionisation potential of the atom.
.P. of H atom = 13.6 V, .P. of He+on= 54.4 V
(v) Excitation Energy :
Energy required to move an electron from ground state of the atom to any other state of the atom is called excitation energy of that state.
Excitation energy of 2ndstate = excitation energy of 1stexcited state = 1stexcitation energy = 10.2 ev.
(vi) Excitation Potential :
Potential difference through which an electron must be accelerated from rest to so that its kinetic energy become equal to excitation energy of any state is called excitation potential of that state.
Excitation potential of third state = excitation potential of second exicited state = second excitation potential = 12.09 V.
(vii) Binding Energy ‘or’ Seperation Energy :
Energy required to move an electron from any state to n = is called binding energy of that state. Binding energy of ground state =.E. of atom or on.
Ex-5 A single electron system has ionization energy 11180 kJ mol–1. Find the number of protons in the nucleus of the system. Sol. .E. = 2 2 n Z × 21.69 × 10–19 J 23 3 10 023 . 6 10 11180 = 2 2 1 Z × 21.69 × 10–19 Ans. Z = 3
HYDROGEN SPECTRUM :
Study of Emission and Absorption Spectra :
An instrument used to separate the radiation of different wavelengths (or frequencies) is called spectroscope or a spectrograph. Photograph (or the pattern) of the emergent radiation recorded on the film is called a spectrogram or simply a spectrum of the given radiation The branch or science dealing with the study of spectra is called spectroscopy.
Spectrum
Based on Nature
Based on origin
Continuous Discrete Absorption
spectrum Emission spectrum Band Spectrum Line Spectrum Emission spectra :
When the radiation emitted from some source e.g. from the sun or by passing electric discharge through a gas at low pressure or by heating some substance to high temperature etc, is passed directly through the prism and then received on the photographic plate, the spectrum obtained is called ‘Emission spectrum’. Depending upon the source of radiation, the emission spectra are mainly of two type :
(a) Continuous spectra :
When white light from any source such as sun, a bulb or any hot glowing body is analysed by passing through a prism it is observed that it splits up into seven different wide band of colours from violet to red. These colours are so continuous that each of them merges into the next. Hence the spectrum is called continuous spectrum.
Figure-14
(b) Discrete spectra : It is of two type (i) Band spectrum
Band Dark space
Figure-15
Band spectrum contains colourful continuous bands sepearted by some dark space. Generally molecular spectrum are band spectrum
(2) Line Spectrum :
Figure-16
This is the ordered arrangement of lines of particular wavelength seperated by dark space eg. hydrogen spectrum.
Line spectrum can be obtained from atoms. 2 . Absorption spectra :
When white light from any source is first passed through the solution or vapours of a chemical substance and then analysed by the spectroscope, it is observed that some dark lines are obtained in the continuous spectrum. These dark lines are supposed to result from the fact that when white light (containing radiations of many wavelengths) is passed through the chemical substance, radiations of certain wavelengths are absorbed, depending upon the nature of the element.
EMISSION SPECTRUM OF HYDROGEN :
Figure-18
When hydrogen gas at low pressure is taken in the discharge tube and the light emitted on passing electric discharge is examined with a spectroscope, the spectrum obtained is called the emission spectrum of hydrogen.
Line Spectrum of Hydrogen :
Line spectrum of hydrogen is observed due to excitation or de-excitation of electron from one stationary orbit to another stationary orbit
Let electron make transition from n2to n1(n2> n1) in a H-like sample
photon n1 n2 eV – 13.6 Z2 n2 2 – 13.6 Z2 n1 2 eV
Energy of emitted photon = (E)n2n1
= 2 2 2 n Z 6 . 13 – 2 1 2 n Z 6 . 13 = 13.6Z2 2 2 2 1 n 1 n 1
Wavelength of emitted photon = 1 2 n n ) E ( hc = 2 2 2 1 2 n 1 n 1 Z 6 . 13 hc 1 = 2 2 2 1 2 n 1 n 1 hc z ) 6 . 13 ( Wave number, 1 = = 2 2 2 1 2 n 1 n 1 RZ R = Rydberg constant = 1.09678 × 107m–1; R~1.1 × 107m–1; R = hc eV 6 . 13 ; R ch = 13.6 eV
Figure-19
Ex-6 Calculate the wavelength of a photon emitted when an electron in H- atom maker a transition from n = 2 to n = 1 Sol. 1 = RZ2 2 2 2 1 n 1 n 1 1 = R(1)2 2 2 2 1 1 1 1 = 4 R 3 or R 3 4
SPECTRA LINES OF HYDROGEN ATOM :
LYMAN SERIES
* It is first spectral series of H.
* It was found out in ultraviolet region in 1898 by Lyman.
* It’s value of n1= 1 and n2= 2,3,4 where ‘n1’ is ground state and ‘n2’ is called excited state of electron present in a H - atom. * 1 = RH 2 2 2 n 1 1 1 where n2> 1 always.
* The wavelength of marginal line =
H 2 1
R n
for all series. So for lyman series=
H
R 1
. * stline of lyman series 2 1
ndline of lyman series = 3 1 Last line of lyman series = 1 [10.2 eV (E)lyman 13.6 eV]
6 . 13 12400 lyman 2 . 10 12400 Aº Aº 12400
* Shortest line : shortest wavelength lineshortestormin= max ) E ( 12400 * First line of any spectral series is the longest (max) line. * Last line of any spectral series is the shortest (min) line. Series limit :
t is the last line of any spectral series. Wave no ofstline of Lyman series
= 1 = = R × 12 2 2 2 1 1 1 = R × 12 4 1 4 = R43 = 4 R 3 R 3 4
Wave no of last line of Lyman series = R × 12 2 2 1 1 1 = R
For Lyman series, longest= 1 2 ) E ( 12400 , shortest =
1 E 12400 BALMER SERIES :* It is the second series of H-spectrum.
* It was found out in 1892 in visible region by Balmer. * It’s value of n1= 2 and n2= 3,4,5,...
* The wavelength of marginal line of Balmer series =
H 2 1 R n = H 2 R 2 = H R 4 * 1 = RH 2 2 2 n 1 2 1 where n2> 2 always.
1.9 (E)balmer 3.4 eV..
All the lines of balmer series in H spectrum are not in the visible range.nfact only st4 lines belongs to visible range.
4 . 3 12400 Aº balmer Å 9 . 1 12400 3648 Å balmer 6536 Å
Lines of balmer series (for H atom) lies in the visible range. Istline of balmer series = 3 2
last line of balmer series = 2 ( ) 1stline = R ×1 2 2 3 1 2 1 = 36 R 5 ( ) last line = R 2 2 1 2 1 = 4 R
PASCHEN SERIES :
(a) It is the third series of H - spectrum.
(b) It was found out in infrared region by Paschen. (c) It’s value of n1= 3 and n2= 4,5,6 ...
(d) The wavelength of marginal line of Paschen series =
H 2 1 R n = H 2 R 3 = H R 9 . (e) 1 = RH 2 2 2 n 1 3 1 where n2> 3 always. BRACKETT SERIES :
(a) It is fourth series of H - spectrum.
(b) It was found out in infrared region by Brackett. (c) It’s value of n1= 4 and n2= 5,6,7 ...
(d) The wavelength of marginal line of brackett series =
H 2 1 R n = H 2 R 4 = H R 16 (e) 1 = RH 2 2 2 n 1 4 1 where n2> 4 always. PFUND SERIES :
(a) It is fifth series of H- spectrum.
(b) It was found out in infrared region by Pfund.
(c) It’s value of n1= 5 and n2= 6,7,8 ... where n1is ground state and n2is excited state.
(d) The wavelength of marginal line of Pfund series =
H 2 1 R n = H 2 R 5 = H R 25 (e) 1 = RH 2 2 2 n 1 5 1 where n2> 5 always. HUMPHRY SERIES :
(a) It is the sixth series of H - spectrum.
(b) It was found out in infrared region by Humphry. (c) It’s value of n1= 6 and n2= 7 , 8 , 9 ... (d) The wavelength of marginal line of Humphry series =
H 2 1 R n = H 2 R 6 = H R 36 (e) 1 = RH 2 2 2 n 1 6 1 where n2> 6.
Ex-7 Calculate wavelength for 2ndline of Balmer series of He+ion
Sol. 2 2 2 1 2 n 1 n 1 ) 2 ( R 1 n1= 2 n2= 4 2 2 2 4 1 2 1 ) 2 ( R 1 4 R 3 1 = 3R 4 Ans.
No. of photons emitted by a sample of H atom :
If an electron is in any higher state n = n and makes a transition to ground state, then total no. of different photons emitted is equal to
2 ) 1 n ( n .
If an electron is in any higher state n = n2and makes a transition to another excited state n = n1, then total no. of different photons emitted is equal to
2 ) 1 n ( n , wheren = n2– n1
Note In case of single isolated atom if electron make transition from nth state to the ground state then max. number of spectral lines observed = (n-1)
Ex-8 If electron make transition from 7th excited state to 2ndstate in H atom sample find the max. number of spectral lines observed.
Sol. n = 8 – 2 = 6
spectral lines = 6 621 = 6× 2 7
= 21
Dual nature of electron (de-Broglie Hypothesis):
(a) Einstein had suggested that light can behave as a wave as well as like a particle i.e. it has dual character. (b) In 1924, de-Broglie proposed that an electron behaves both as a material particle and as a wave. (c) This proposed a new theory wave mechanical theory of matter. According to this theory, the electrons protons and even atom when in motion possess wave properties.
(d) According to de-Broglie, the wavelength associated with a particle of mass m, moving with velocity v is given by the relation,
= mv
h
where h is Planck’s constant
(e) This can be derived as follows according to Planck’s equation. E = h = h.c
Energy of photon on the basis of Einstein’s mass energy relationship
E = mc2 or =
mc h Equating both we get
c . h = mc2 or = mc h
Which is same as de - Broglie relation.
This was experimentally verified by Davisson and Germer by observing diffraction effects with an electron beam.
Let the electron is accelerated with a potential of V then the K.E. is
2 1
mv2=eV
m2v2= 2emV
=
emV 2
h
If we associate Bohr’s theory with de - Broglie equation then 2r = n or =
n r 2 From de-Broglie equation = mv h therefore mv h = n r 2 so, mvr = 2 nh m = dynamic mass = 2 0 c v 1 m
m0= rest mass of particle depended on velocity c = velocity of light
If velocity of particle is zero then : dynamic mass = rest mass
Rest mass of photon is zero that means photon is never at rest * K.E. = 2 1 mv2 m (K.E.) = 2 1
m2v2multiplied by mass on both side m.v. = 2m(K.E.) = .) E . K ( m 2 h
If a charge q is accelerated through a potential difference of ‘V’ volt from rest then K.E. of the charge is equal to : “ q.V” = ) V . q ( m 2 h
* If an electron is accelerated through a potential difference of ‘V’ volt from rest then :
= ) eV ( m 2 h e = 2 1 V 150
Å (on putting values of h, meand e)
= V 3 . 12 Å (V in volt) * mvr = n × 2 h = mv h
mv = h putting this in mvr = 2 nh hr = 2 nh 2nr de Broglie wavelength
Ex-9 What will be the wavelength of a ball of mass 0.1 kg moving with a velocity of 10 m s–1? (NCERT) Sol. According to de Broglie equation
= mv h = ) s m 10 ( ) kg 1 . 0 ( ) Js 10 626 . 6 ( 1 – 34 – = 6.626 × 10–34m (J = kg m2s–2).
Heisenberg’s Uncertainty Principle :
The exact position and momentum of a fast moving particle cannot be calculated precisely at the same moment of time. Ifx is the error in the measurement of position of the particle and if p is the error in the measurement of momentum of the particle, then:
x . p 4 h or x . (mv) 4 h where, x = uncertainty in position
p = uncertainty in momentum h = Plank’s constant
m = mass of the particle v = uncertainty in velocity
If the position of a particle is measured precisely, i.e.x 0 then p . If the momentum of the particle is measured precisely. i.e.p 0 then x .
This is because of a principle of optics that if a light of wavelength ‘’ is used to locate the position of a particle then maximum error in the position measurement will be
i.e.x = Ifx 0 ; 0
But, p = h p
So, to makex 0 , 0 a photon of very high energy is used to locate it.
When this photon will collide with the electron then momentum of electron will get changed by a large amount.
* p.x 4
h
(multiplied & divided byt)
x . t t P 4 h ( t P
= rate of change in momentum = F) F.x.t 4h
E . t
4 h
E uncertainty in energy t uncertainty in time
In terms of uncertainty in energyE, and uncertainty in time t, this principle is written as, E.t 4 h .
Heisenberg replaced the concept of definite orbits by the concept of probability.Ex-10 A golf ball has a mass of 40 g, and a speed of 45 m/s. If the speed can be measured within accuracy of 2%, calculate the uncertainty in the position. (NCERT) Sol. The uncertainty in the speed is 2%, i.e., 45 ×
100 2
= 0.9 m s–1.
Using the equation x = v m 4 h = 4 3.14 40 10 (0.9ms ) 10 626 . 6 1 – 3 – 34 – = 1.46 × 10–33m
This is nearly ~ 1018times smaller than the diameter of a typical atomic nucleus. As mentioned earlier for
large particles, the uncertainty principle sets no meaningful limit to the precision of measurements.
Orbital :
An orbital may be defined as the region of space around the nucleus where the probability of finding an electron is maximum (90% to 95%)
Orbitals do not define a definite path for the electron, rather they define only the probability of the electron being in various regions of space around the nucles.
Figure-20
Difference between orbit and orbitals
Orbit
Orbitals
1. It is well defined circular path followed 1. It is the region of space around the nucleus by revolving electrons around the nucleus where electron is most likely to be found 2. It represents planar motion of electron 2. It represents 3 dimensional motion of
an electron around the nucleus.
3. The maximum no. of electron in an 3. Orbitals can not accomodate more than orbits is 2n2where n stands for no. 2 electrons.
of orbit.
4. Orbits are circular in shape. 4. Orbitals have different shape e.g. s-orbital is spherical, p - orbital is dumb- bell shaped.
5. Orbit are non directional in character. 5. Orbitals (except s-orbital) have directional character. Hence, they cannot explain shape of Hence, they can account for the shape of molecules. molecules
6. Concept of well defined orbit is against 6. Concept of orbitals is in accordance with Heisenberg’s Heisenberg’s uncentainty principle. principle
18.
Shape of the orbitals :
Shape of the orbitals are related to the solutions of Schrodinger wave equation, and gives the space in which the probability of finding an electron is maximum.
s- orbital :
Shape
spherical
Figure-21
s- orbital is non directional and it is closest to the nucleus, having lowest energy. s-orbital can accomodate maximum no. of two electrons.
p-orbital : Shape
dumb bell
Dumb bell shape consists of two lobes which are separated by a region of zero probability called node.
Figure-22
p - orbital can accomodate maximum no. of six electrons.Figure-23
d - orbital can accomodate maximum no. of 10 electrons.f - orbital :
Shape
leaf like
f - orbital can accomodate maximum no. of 14 electrons.
QUANTUM NUMBERS :
The set of four numbers required to define an electron completely in an atom are called quantum numbers. The first three have been derived from Schrodinger wave equation.
(i) Principal quantum number (n) : (Proposed by Bohr)
It describes the size of the electron wave and the total energy of the electron. It has integral values 1, 2, 3, 4 ...., etc., and is denoted by K, L, M, N. ..., etc.
* Number of subshell present in nthshell = n
n subshell
1 s
2 s, p
3 s, p, d
4 s, p, d, f
* Number of orbitals present in nthshell = n2.
* The maximum number of electrons which can be present in a principal energy shell is equal to 2n2.
No energy shell in the atoms of known elements possesses more than 32 electrons. * Angular momentum of any orbit =
2 nh
(ii) Azimuthal quantum number () : (Proposed by Sommerfield)
It describes the shape of electron cloud and the number of subshells in a shell. * It can have values from 0 to (n – 1)
* value of subshell
0 s
1 p
2 d
3 f
* Number of orbitals in a subshell = 2 + 1
* Maximum number of electrons in particular subshell = 2 × (2 + 1) * Orbital angular momentum L =
2 h ) 1 ( = (1) 2h
i.e. Orbital angular momentum of s orbital = 0, Orbital angular momentum of p orbital = 2
h 2 ,
Orbital angular momentum of d orbital = 2
h 3
(iii) Magnetic quantum number (m) : (Proposed by Linde)
It describes the orientations of the subshells. It can have values from – to + including zero, i.e., total (2 + 1) values. Each value corresponds to an orbital. s-subshell has one orbital, p-subshell three
orbitals (p
x, pyand pz), d-subshell five orbitals (dxy,dyz,dzx,dx2y2,dz2) and f-subshell has seven orbitals.The total number of orbitals present in a main energy level is ‘n2’.
(iv) Spin quantum number (s) : (Proposed by Samuel Goldsmit & Uhlenbeck)
It describes the spin of the electron. It has values +1/2 and –1/2. signifies clockwise spinning and anticlockwise spinning.
* Spin magnetic moments= 2 mc eh
s(s1) or
= n(n2) B.M. (n = no. of unpaired electrons) * It represents the value of spin angular momentum which is equal to 2 h ) 1 s ( s
* Maximum spin of atom = 2 1
x No. of unpaired electron.
Electronic configuration :
Pauli’s exclusion principle :
No two electrons in an atom can have the same set of all the four quantum numbers, i.e., an orbital cannot have more than 2 electrons because three quantum numbers (principal, azimuthal and magnetic) at the most may be same but the fourth must be different, i.e., spins must be in opposite directions.
Aufbau principle :
Aufbau is a German word meaning building up. The electrons are filled in various orbitals in order of their increasing energies. An orbital of lowest energy is filled first. The sequence of orbitals in order of their increasing energy is :
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, .... The energy of the orbitals is governed by (n +) rule.
‘n +Rule :
The relative order of energies of various sub-shell in a multi electron atom can be predicated with the help of ‘n +’ rule
The sub-shell with lower value of (n + ) has lower energy and it should be filled first.
eg. 3d 4s
(n +) = 3 + 2 (n +) = 4 + 0
= 5 = 4
Since, (n +) value of 3d is more than 4s therefore, 4s will be filled before 3d.
If two sub-shell has same value of (n + ) then the sub-shell with lower value of n has lower energy and it should be filled first.
eg. 3d 4p
(n +) = 3 + 2 = 4 + 1
= 5 = 5
MEMORY MAP : 1s 2s 3s 4s 5s 6s 2p 3p 4p 5p 6p 3d 4d 4f 5d 5f 1 – s 2 3 – s,p 4 – s,p – s,d,p 5 – s,d,p 6 – s,f,d,p 6 – s,f,d,p 7 – s,f,d,p
Hund’s rule :
No electron pairing takes place in the orbitals in a sub - shell until each orbital is occupied by
one electron with parallel spin. Exactly half filled and fully filled orbitals make the atoms more stable, i.e., p3,
p6, d5, d10, f7and f14configuration are most stable.
Ex-11 Write the electronic configuration and find the no. of unpaired electrons as well as total spin for the following atoms : (1) 6C (2) 8O (3) 15P (4) 21Sc (5) 26Fe (6) 10Ne Sol. (i) 6C 1s2, 2s2, 2p2
No. of unpaired electrons 2. Total spin = 2 2 or 2 2 (ii) 8O 1s2, 2s2, 2p4 1s 2s 2p
No. of unpaired electrons = 2 Total spin = 2 2 or 2 2 (iii) 15P1s2, 2s2, 2p6, 3s2, 3p3 3s 3p
No. of unpaired electrons = 3 Total spin = 2 3 or 2 3
(iv) 21Sc 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d1 or [Ar] 4s23d1
3d 4s
[Ar] 3d14s2
No. of unpaired electrons = 1 Total spin = 2 1 or 2 1 (v) 26Fe 1s2, 2s2 2p6, 3s2, 3p6, 4s23d6 or [Ar] 4s2, 3d6 3d 4s
No. of unpaired electrons = 4 Total spin = 2 4 or 2 4 (vi) 10Ne 1s2, 2s22p6
No. of unpaired electrons = 0 Total spin = 0
Ex-12 Write down the four quantum numbers for fifth and sixth electrons of carbon atom. Sol. 6C : 1s2, 2s22p2 fifth electron : n = 2 = 1 m = – 1 or +1 s = + 2 1 or – 2 1 sixth electron : n = 2 = 1 m = 0 s = + 2 1 or – 2 1
Ex-13 Calculate total spin, magnetic moment for the atoms having at. no. 7, 24 and 36. Sol. The electronic configuration are
7N : 1s2, 2s22p3 unpaired electron = 3
24Cr : 1s2, 2s22p6, 3s23p63d5, 4s1 unpaired electron = 6 36Kr : 1s2, 2s22p6, 3s23p63d10, 4s24p6 unpaired electron = 0 Total spin for an atom = ± 1/2 × no. of unpaired electron
For7N, it is = ± 3/2 ; For24Cr, it is = ± 3 ; For36Kr, it is = 0 Also magnetic moment = n(n2)
For7N, it is = 15 ; For24Cr, it is = 48 ; For36Kr, it is = 0
EXCEPTIONS :
(1) 24Cr = [Ar] 4s2, 3d4(Not correct)
[Ar] 4s1, 3d5 (correct : as d5structure is more stable than d4structure) (2) 29Cu = [Ar] 4s1, 3d10(correct : as d10structure is more stable than d9structure).
THE SCHRODINGER EQUATION :
The de Broglie wave relation is the basis for predicting the behavior of freely moving particles. Shortly after it was proposed, Erwin Schrodinger demonstrated that the de Broglie expression could be generalized so as to apply to bound particles such as electrons in atoms. The heart of Schrodinger's theory is that the allowed energies of physical systems can be found by solving an equation which so resembles the equations of classical wave theory that it is called the wave equation. For the motion of one particle in one (the x) direction, the Schrodinger wave equation is
– m 8 h 2 2 2 2 dx d + V = E .
The "knowns" in this equation are m, the mass of the particle, and V, its potential energy expressed as a function of x. The "unknowns" to be found by solving the equation are E, the quantized or allowed energies of the particle, and, which is called the wave function. The quantity d2/dx2represents the rate of change of
d/dx, the rate of change of . When this equation is applied to real systems such as the hydrogen atom, it is found that it cannot be solved unless E takes on certain values which are related by integers. Thus quantized energy and quantum numbers are an automatic consequence of the Schrodinger theory, and do not have to be tacked on to Newtonian mechanics as was done by Bohr.
What is ? By itself, it has no physical meaning. However, the square of the absolute value of , 2, does
have an important physical interpretation. It is a mathematical expression of how the probability of finding a particle varies from place to place. Thus the exact trajectories of Newtonian mechanics and the Bohr theory do not appear in the results of the Schrodinger quantum mechanics; this, according to the Uncertainty Principle, is as it should be.
A similar analysis is possible for the other 2p-functions. The p
xfunction has the yz-plane as an angular node, since the function is proportional sin cos and cos = 0 everywhere in the yz-plane. The maximum values of 1 for sin and cos occur along the positive x-axis. The py function, proportional to the sin sin , vanishes in the xz-plane, where sin = 0, and has a maximum along the positive y-axis, where both sin and sin are unity.FIG. The 2p-orbitals of the hydrogen atom. (Adapted from K. B. Harvey and G. B. Porter, An Introduction to Physical Inorganic Chemistry. Reading, Mass.: Addison-Wesley, 1963.)
The 3d-orbitals of the hydrogen atom. Note the relation between the labeling of the d-orbitals and their orientations in space. (Adapted from K. B. Harvey and G. B. Porter, An Introduction to Physical Inorganic Chemistry. Reading,
Radial probability density for some orbitais of the hydrogen atom. Ordinate is proportional to 4 r2R2, and all distributions are to the same scale.
(i) total nodes = n – 1, (ii) angular nodes =, (iii) radial nodes = n – – 1.
MISCELLANEOUS SOLVED PROBLEMS (MSPs)
1. The ratio of (E2– E1) to (E4– E3) for He+ion is approximately equal to (where Enis the energy of n thorbit)
Sol. 2 2 2 2 2 2 ) 4 ( 1 ) 3 ( 1 ) 2 ( 6 . 13 ) 2 ( 1 ) 1 ( 1 ) 2 ( 6 . 13 = 15
2. If the binding energy of 2ndexcited state of a hydrogen like sample is 24 eV approximately, then the ionisation energy of the sample is approximately
(A) 54.4 eV (B) 24 eV (C) 122.4 eV (D*) 216 eV Sol. 2 2 ) 3 ( ) Z ( 6 . 13 = 24 I.E. = 13.6(Z)2= (24 × 9) = 216 ev
3. The ionisation energy of H atom is 21.79 × 10–19J . Then the value of binding energy of second excited state of Li2+ion (A) 32× 21.7 × 10–19J (B*) 21.79 × 10–19J (C) 3 1 × 21.79 × 10–19J (D) 2 3 1 × 21.79 × 10–19J Sol. B.E. = 2 2 19 ) 3 ( ) 3 ( 10 79 . 21 = 21.79 × 10–19J
4. The wave number of the first line in the Balmer series of hydrogen is 15200cm1. What would be the wavenumber of the first line in the Lyman series of the Be3+ion?
(A) 2.4 x 105cm1 (B) 24.3 x 105cm1 (C) 6.08 x 105cm1 (D*) 1.313 x 106cm1 Sol. Given 15200 = R(1)2 2 2 ) 3 ( 1 ) 2 ( 1 ... (1) Then = R(4)2 2 2 (2) 1 ) 1 ( 1 ... (2) from (1) and (2) equation
= 1.313 × 106cm–1
5. What would be the maximum number of emission lines for atomic hydrogen that you would expect to see with the naked eye if the only electronic energy levels involved are those as shown in the Figure?
Hint : Balmer series lines lies in visible region.
(A*) 4 (B) 6 (C) 5 (D) 15
Sol. Only four lines are present in visible region, 6 2, 5 2, 4 2 & 3 2.
6. The de Brogile wavelength of an electron moving in a circular orbit is. The minimum radius of orbit is
(A) (B*) 2 (C) 4 (D) 3
Sol. We know 2r = n
For minimum radius n = 1 2rmin=
rmin=
2
7. An electron, practically at rest, is initially accelerated through a potential difference of 100 volts. It then has a de Broglie wavelength =1Å. It then get retarded through 19 volts and then has a wavelength2Å. A further retardation through 32 volts changes the wavelength to3, What is
1 2 3 ? (A) 41 20 (B) 63 10 (C*) 63 20 (D) 41 10 Sol. 1= 100 150 Å .... (1) 2= 81 150 Å .... (2) 3= 49 150 Å .... (3) From (1), (2) and (3) 63 20 1 2 3
8. Uncertainty in position of a hypothetical subatomic particle is 1Å and uncertainty in velocity is
4 3 . 3
× 105m/ s then the mass of the particle is approximately [h = 6.6 × 10–34Js]
(A) 2 × 10–28kg (B) 2 × 10–27kg (C*) 2 × 10–29kg (D) 4 × 10–29kg Sol. x × m × v h/4 1 × 10-10× m × 4 3 . 3 × 105 4 10 6 . 6 34 m = 2 × 10– 29kg 9. Which of the following set of quantum numbers is not valid.
(A) n = 3, = 2, m = 2, s = + 2 1 (B) n = 2, = 0, m = 0, s = – 2 1 (C) n = 4, = 2, m = –1, s = + 2 1 (D*) n = 4, = 3, m = 4, s = – 2 1
Sol. (D) Not valid
10. What is the total spin value in case of26Fe3+ion
(A) +1 or –1 (B) +2 or –2 (C*) + 2.5 or – 2.5 (D) +3 or –3
Sol. Total spin = no. of unpaired e–×
2 1 = 5 × 2 1 = ± 2 5
